Question

2. Let X1, ..., Xn be a random sample from a uniform distribution on the interval (0, θ) where θ > 0 is a parameter. The prior distribution of the parameter has the pdf f(t) = βαβ/t^(β−1) for α < t < ∞ and zero elsewhere, where α > 0, β > 0. Find the Bayes estimator for θ. Describe the usefulness and the importance of Bayesian estimation.

We are assuming that theta = t, but we are unsure if that would make the problem workable.

Answer 1

you are assuming correct.. because we use bayesian analysis when our atleast one parameter of distribution of sample is random. here our sample is from randomly taken from uniform(0,theta), but here we assume that theta is unkniwn but random and prior distribution of theta ispdf f(theta) = βα^β/theta^(β−1) for α < t < ∞, therefore we can find posterior distribution such as

which is the pdf of pareto distribution. now the bayes estimator for theta is the mean of pareto distribution which is now if we discuss about the usefullness and importance of Bayesian estimation so we can say that we havi discuss that in classical inference we assume that parameter are constant while there is many time situations occur where our parameter behave a r.v than bayesian inference give better result as compared to classical inference and 2nd thing is it give good estimate of population parameters with the help of combined information of sample and prior distribution of parameters.

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